A note on rings with the summand sum property

Abstract

A ring $R$ is called right SSP (SIP) if the sum (intersection) of any two direct summands of $R_{R}$ is also a direct summand. Left sides can be defined similarly. The following are equivalent: (1) $R$ is right SSP. (2) $R$ is right C3 and right SIP. (3) $R$ is left C3 and left SIP. (4) $R$ is left SSP. It is also shown that (1) $R$ is a von-Neumann regular ring if and only if $\mathbb{M}_{2}(R)$ is right SSP if and only if $\mathbb{M}_{n}(R)$ is right SSP for some $n>1$; (2) $R$ is a semisimple ring if and only if the column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\Lambda}(R)$ is right SSP for a countably infinite set $\Lambda$ if and only if the column finite matrix ring $\mathbb{C}\mathbb{F}\mathbb{M}_{\Lambda}(R)$ is right SSP for any infinite set $\Lambda$. Some known results are improved.